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He went to work by his car at speed 120 km/hr and back home the same way 90 km/hr, find the average speed for the entire distance traveled.

$${He}\:{went}\:{to}\:{work}\:{by}\:{his}\:{car}\:{at}\:{speed}\:\mathrm{120}\:{km}/{hr} \\ $$$$\:{and}\:{back}\:{home}\:{the}\:{same}\:{way}\:\mathrm{90}\:{km}/{hr},\:{find}\:{the} \\ $$$$\:{average}\:{speed}\:{for}\:{the}\:{entire}\:{distance}\:{traveled}. \\ $$$$ \\ $$

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A spring with length L and spring constant k is fixed on the ceiling. Hang a mass point m on the bottom of the spring. Find the relation between distsnce h from m to ceiling and time t. Ignore air resistance, friction and gravity of spring.

$$\mathrm{A}\:\mathrm{spring}\:\mathrm{with}\:\mathrm{length}\:{L}\:\mathrm{and}\:\mathrm{spring}\:\mathrm{constant}\:{k} \\ $$$$\mathrm{is}\:\mathrm{fixed}\:\mathrm{on}\:\mathrm{the}\:\mathrm{ceiling}.\:\mathrm{Hang}\:\mathrm{a}\:\mathrm{mass}\:\mathrm{point}\:{m} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{bottom}\:\mathrm{of}\:\mathrm{the}\:\mathrm{spring}.\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{relation}\:\mathrm{between}\:\mathrm{distsnce}\:{h}\:\mathrm{from}\:{m}\:\mathrm{to}\:\mathrm{ceiling} \\ $$$$\mathrm{and}\:\mathrm{time}\:{t}.\:\mathrm{Ignore}\:\mathrm{air}\:\mathrm{resistance},\:\mathrm{friction}\:\mathrm{and} \\ $$$$\mathrm{gravity}\:\mathrm{of}\:\mathrm{spring}. \\ $$

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Zombie apocalypse has started. You are at point A (3,2). At point B (−5,4) there is a shelter. A river flows along the X axis. You are running at a constant velocity 1 ms^(−1) with an intention to reach the shelter but before that, you have to reach the river and fill-up your water jar. What is the lowest time it takes to reach B from A? [It takes 1 second time to fill the jar with water.] a) 11.0 s b) 8.24 s c) 10.1 s d) 9 s

$${Zombie}\:{apocalypse}\:{has}\:{started}.\:{You} \\ $$$${are}\:{at}\:{point}\:{A}\:\left(\mathrm{3},\mathrm{2}\right).\:{At}\:{point}\:{B}\:\left(−\mathrm{5},\mathrm{4}\right) \\ $$$${there}\:{is}\:{a}\:{shelter}.\:{A}\:{river}\:{flows}\:{along}\: \\ $$$${the}\:{X}\:{axis}.\:{You}\:{are}\:{running}\:{at}\:{a}\:{constant} \\ $$$${velocity}\:\mathrm{1}\:{ms}^{−\mathrm{1}} \:{with}\:{an}\:{intention}\:{to} \\ $$$${reach}\:{the}\:{shelter}\:{but}\:{before}\:{that},\:{you} \\ $$$${have}\:{to}\:{reach}\:{the}\:{river}\:{and}\:{fill}-{up}\:{your} \\ $$$${water}\:{jar}.\:{What}\:{is}\:{the}\:{lowest}\:{time}\:{it}\: \\ $$$${takes}\:{to}\:{reach}\:{B}\:{from}\:{A}?\:\left[{It}\:{takes}\:\mathrm{1}\:{second}\right. \\ $$$$\left.{time}\:{to}\:{fill}\:{the}\:{jar}\:{with}\:{water}.\right]\: \\ $$$$ \\ $$$$\left.{a}\right)\:\mathrm{11}.\mathrm{0}\:{s} \\ $$$$\left.{b}\right)\:\mathrm{8}.\mathrm{24}\:{s} \\ $$$$\left.{c}\right)\:\mathrm{10}.\mathrm{1}\:{s} \\ $$$$\left.{d}\right)\:\mathrm{9}\:{s} \\ $$

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An open pipe of r_o = 10 Cm, ℓ= 3m has an outer layer of ice that is melting at the rate of 2π Cm^3 per minute with thickness of 20 mm. How many days untill all the ice melts? and how fast is the thickness of the ice decreasing per hour?

$${An}\:{open}\:{pipe}\:{of}\:{r}_{{o}} =\:\mathrm{10}\:{Cm},\:\ell=\:\mathrm{3}{m}\:{has}\:{an}\:{outer} \\ $$$$\:{layer}\:{of}\:{ice}\:{that}\:{is}\:{melting}\:{at}\:{the}\:{rate}\:{of}\:\mathrm{2}\pi\:{Cm}^{\mathrm{3}} \\ $$$$\:{per}\:{minute}\:{with}\:{thickness}\:{of}\:\mathrm{20}\:{mm}.\:{How}\:{many} \\ $$$$\:{days}\:{untill}\:{all}\:{the}\:{ice}\:{melts}?\:{and}\:{how}\:{fast}\:{is}\:{the} \\ $$$$\:{thickness}\:{of}\:{the}\:{ice}\:{decreasing}\:{per}\:{hour}? \\ $$

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A projectile is fired with velocity(v_o ) such that it passes through two points both a distance(h) above the horizontal.show that if the gun is adjusted for the maximum range of the separation of two position is d=((v_o (√(v_o ^2 −4gh)))/g)

$$\mathrm{A}\:\mathrm{projectile}\:\mathrm{is}\:\mathrm{fired}\:\mathrm{with}\:\mathrm{velocity}\left(\mathrm{v}_{\mathrm{o}} \right) \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{it}\:\mathrm{passes}\:\mathrm{through}\:\mathrm{two} \\ $$$$\mathrm{points}\:\mathrm{both}\:\mathrm{a}\:\mathrm{distance}\left(\mathrm{h}\right)\:\mathrm{above}\:\mathrm{the} \\ $$$$\mathrm{horizontal}.\mathrm{show}\:\mathrm{that}\:\mathrm{if}\:\mathrm{the}\:\mathrm{gun}\:\mathrm{is}\:\mathrm{adjusted} \\ $$$$\mathrm{for}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{range}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{separation}\:\mathrm{of}\:\mathrm{two}\:\mathrm{position}\:\mathrm{is} \\ $$$$\boldsymbol{\mathrm{d}}=\frac{\boldsymbol{\mathrm{v}}_{\boldsymbol{\mathrm{o}}} \sqrt{\boldsymbol{\mathrm{v}}_{\boldsymbol{\mathrm{o}}} ^{\mathrm{2}} −\mathrm{4}\boldsymbol{\mathrm{gh}}}}{\boldsymbol{\mathrm{g}}} \\ $$

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A particle is projected vertically upward in a constant gravitation field with initial speed (v_0 ).show that there is retarding proportional to the square of the instantaneous speed,the speed of the partical when it returns on the initial position is ((v_o v_c )/( (√(v_o ^2 +v_c ^2 )))) where v_c is terminal speed

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{is}\:\mathrm{projected}\:\mathrm{vertically} \\ $$$$\mathrm{upward}\:\mathrm{in}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{gravitation} \\ $$$$\mathrm{field}\:\mathrm{with}\:\mathrm{initial}\:\mathrm{speed}\:\left(\mathrm{v}_{\mathrm{0}} \right).\mathrm{show} \\ $$$$\mathrm{that}\:\mathrm{there}\:\mathrm{is}\:\mathrm{retarding}\:\mathrm{proportional} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{square}\:\mathrm{of}\:\mathrm{the}\:\mathrm{instantaneous} \\ $$$$\mathrm{speed},\mathrm{the}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{the}\:\mathrm{partical} \\ $$$$\mathrm{when}\:\mathrm{it}\:\mathrm{returns}\:\mathrm{on}\:\mathrm{the}\:\mathrm{initial} \\ $$$$\mathrm{position}\:\mathrm{is}\:\:\:\frac{\boldsymbol{\mathrm{v}}_{\boldsymbol{\mathrm{o}}} \boldsymbol{\mathrm{v}}_{\boldsymbol{\mathrm{c}}} }{\:\sqrt{\boldsymbol{\mathrm{v}}_{\boldsymbol{\mathrm{o}}} ^{\mathrm{2}} +\boldsymbol{\mathrm{v}}_{\boldsymbol{\mathrm{c}}} ^{\mathrm{2}} }}\:\:\boldsymbol{\mathrm{where}}\:\boldsymbol{\mathrm{v}}_{\boldsymbol{\mathrm{c}}} \:\boldsymbol{\mathrm{i}}\mathrm{s} \\ $$$$\mathrm{terminal}\:\mathrm{speed} \\ $$

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Imagine that a hole is drilled from one point on the earth surface to the other side through the diameter of the earth and small of mass ^′ m^′ released into the hole. show that the hole of mass excute simple harmonic motion and find its period of revolution

$$\mathrm{Imagine}\:\mathrm{that}\:\mathrm{a}\:\mathrm{hole}\:\mathrm{is}\:\mathrm{drilled} \\ $$$$\mathrm{from}\:\mathrm{one}\:\mathrm{point}\:\mathrm{on}\:\mathrm{the}\:\mathrm{earth}\: \\ $$$$\mathrm{surface}\:\mathrm{to}\:\mathrm{the}\:\mathrm{other}\:\mathrm{side}\:\mathrm{through}\:\mathrm{the} \\ $$$$\mathrm{diameter}\:\mathrm{of}\:\mathrm{the}\:\mathrm{earth}\:\mathrm{and}\:\mathrm{small} \\ $$$$\mathrm{of}\:\mathrm{mass}\:\:^{'} \mathrm{m}^{'} \:\mathrm{released}\:\mathrm{into}\:\mathrm{the}\:\mathrm{hole}. \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{hole}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{excute} \\ $$$$\mathrm{simple}\:\mathrm{harmonic}\:\mathrm{motion}\:\mathrm{and} \\ $$$$\mathrm{find}\:\mathrm{its}\:\mathrm{period}\:\mathrm{of}\:\mathrm{revolution} \\ $$

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